论文标题
张量排名的概念
Notions of Tensor Rank
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.
